Research Article | Open Access
Well Posedness for a Class of Flexible Structure in Hölder Spaces
We characterize well-posedness in Hölder spaces for an abstract version of the equation which model the vibrations of flexible structures possessing internal material damping and external force . As a consequence, we show that in case of the Laplacian with Dirichlet boundary conditions, equation is always well-posed provided .
During the last few decades, the use of flexible structural systems has steadily increased importance. The study of a flexible aerospace structure is a problem of dynamical system theory governed by partial differential equations.
We consider here the problem of characterize well posedness, for a mathematical model of a flexible space structure like a thin uniform rectangular panel. For example, a solar cell array or a spacecraft with flexible attachments. This problem is motivated by both engineering and mathematical considerations.
Such mechanical system was mathematically introduced in  and consists of a short rigid hub, connected to a flexible panel of length . Control torque is applied to the hub. The panel is made of viscoelastic material with internal Voigt-type damping with coefficient , that is, an ideal dashpot damping which is directly proportional to the first derivative of the longitudinal displacement, and opposing the direction of motion. The equation of motion of the panel is given by where is the velocity of longitudinal wave propagation, , and are, respectively, torsional rigidity, density and radius of gyration about the central axis of the panel. Initial position and deflection angle are known. In  exact controllability and boundary stabilization for the solution of (1.1) was analyzed and in [2, p. 188], the exact decay rate was obtained.
In  the explicit exponential energy decay rate was obtained for the solution of (1.2) subject to mixed boundary conditions. However, consideration of external forces interacting with the system, which lead us naturally with the well posedness for the nonhomogeneous version of (1.2), appears as an open problem.
In the first part of this paper we study well posedness of the following abstract version of (1.2):
where is a closed linear operator acting in a Banach space and is a -valued function. We emphasize that when in general one cannot expect that (1.3) is well posed due to the presence of the term . In fact, it is well known that the abstract Cauchy problem associated with (1.3) is in general ill posed, see for example .
We are able to characterize well posedness, that is, temporal maximal regularity, of solutions of (1.3) solely in terms of boundedness of the resolvent set of . This will be achieved in the Hölder spaces where The methods to obtain this goal are those incorporated in  where a similar problem in case of the first order abstract Cauchy problem has been studied.
Let be Banach spaces, we write for the space of bounded linear operators from to and let We denote by the spaces normed by Let be an open set. By we denote the space of all -functions in having compact support in .
We denote by or the Fourier transform, that is,
Definition 2.1. Let be continuous. We say that is a -multiplier in if there exists a mapping such that for all and all
Here Note that is well defined, linear and continuous (cf. [6, Definition 5.2]).
Define the space as the set with the norm Let (where is a positive integer) be the Banach space of all such that , equipped with the norm Observe from Definition 2.1 and the relation that for we have . Moreover, if is bounded then is bounded as well (see [6, Remark 6.3]). The following multiplier theorem is due to Arendt, Batty and Bu [6, Theorem 5.3].
Theorem 2.2. Let be such that Then M is a -multiplier.
3. A Characterization of Well Posedness in Hölder Spaces
In this section we characterize -well posedness. Given we consider in this section the linear problem
where is a closed linear operator in and Note that the solution of (3.1) does not have to satisfy any initial condition. In the case , solutions of (3.1) with periodic boundary conditions has been recently studied in . On the other hand, well posedness of the homogeneous abstract Cauchy problem has been observed recently in  for and all under certain assumptions on . See also  for related maximal regularity results in the case of a damped wave equation.
We denote by the domain of considered as a Banach space with the graph norm.
In the next proposition, as usual we denote by the resolvent set and resolvent of the operator , respectively.
Proposition 3.2. Assume that (3.1) is -well-posed. Then(i) for all and,(ii)
Proof. Denote by the bounded operator which associates to each the unique solution of (3.1). Let Let be such that Define Then it is not difficult to see that is a solution of (3.1) with Hence, by uniqueness,
Let and define Let For fixed we define Then is easy to check that and are both solutions of (3.1) with replaced by By uniqueness, for all In particular, it follows that for all Let Replacing in (3.1) we obtain Taking we conclude that is bijective and Define and We have the identity where (see [6, section 3]). Hence Therefore, for we have On the other hand, since is bounded and is continuous at , we obtain (ii) and the proof is complete.
In what follows, we denote by the function: for all and As before, we also use the notation
Lemma 3.3. Assume that then and are - multipliers in Moreover and are -multipliers in
Proof. Define . We first observe that the functions and have the property that and are bounded on We next claim that is a -multiplier. In fact, note that by hypothesis for each and the function is continuous at since Hence is bounded. Moreover, defining we have where is of order and then is bounded by (3.8). It follows that is bounded. Next, we have the identity where the first three terms on the right hand side are bounded. For the last term, we have It is clear that the first two terms on the right hand side are bounded. We observe that the last term also is bounded. In fact, note that by hypothesis for each and the function is continuous at Hence is bounded. This completes the proof of the Lemma.
Lemma 3.4. Let and The following assertions are equivalent:(i) and is constant.(ii) for all
Proof. . Let . Then
. Let and . Then and . Let . Then integration by parts and assumption give . It follows from [10, Theorems 4.8.2 and 4.8.1] that is a polynomial. Since it follows that for some vectors . Thus for some vector
The following theorem, which is one of the main results in this paper, shows that the converse of Proposition 3.2 is valid.
Theorem 3.5. Let be a closed linear operator defined on a Banach space . Then the following assertions are equivalent:(i)Equation (3.1) is -well posed;(ii) for all and
Proof. The implication follows by Proposition 3.2. We now prove the converse implication.
Let By Lemma 3.3 there exists and such that for all . Choosing in (3.13), it follows from Lemma 3.4 that and for some Now we can choose in (3.14), it follows that and for some In a similar way, we can see that and for some From the definition of we obtain . Taking into account the definition of we get Then we deduce the identity We multiply the above identity by , take Fourier transforms and then integrate over after taking the values at , we obtain for all Using (3.17), (3.18) and (3.19) in the above identity we conclude that for all By Lemma 3.4 there exists such that We define Then, we can show that solves (3.1) and that
In order to prove uniqueness, suppose that where Let We define by where the hat indicates the Carleman transform (see e.g. ). By [12, Proposition ], we have that for all the Schwartz space of smooth rapidly decreasing functions on We will prove that the right term in (3.27) is zero, from which proving the theorem. In fact, by [12, Proposition ] we have where Observe that for all Therefore we have Let Multiplying by and integrating over the above identity we obtain where We note that in [12, Lemma A.4], It remains to prove that In fact, since , we have Then It is easy to check that as proving (3.34).
We write We first prove that In fact, we apply Fubini's theorem to obtain It follows from [, Lemma A.3] that where is a positive constant. Taking into account (3.39) and (3.40) we deduce (3.38).
We next prove that In fact, define Then By [12, Lemma ], we have for where Therefore, we deduce (3.41). Proceeding in the same way we obtain This completes the proof of the assertion (3.34).
The following consequence of Theorem 3.5 is remarkable in the study of well posedness for flexible structural systems. We recall that
Corollary 3.7. If is the generator of a bounded analytic semigroup, then (3.1) is -well posed.
Proof. Since generates a bounded analytic semigroup, we have that and there is a constant such that for Note that and that We conclude that and The conclusion follows by Theorem 3.5.
For example, if is a normal operator on a Hilbert space satisfying for some then generates a bounded analytic semigroup. In particular, the semigroup generated by a self-adjoint operator that is bounded above is analytic of angle Another important class of generators of analytic semigroups is provided by squares of group generators.
Example 3.8. Since the Laplacian is the generator a bounded analytic semigroup (the diffusion semigroup) in we obtain that for each the problem has a unique solution
Since it is also well known that the Dirichlet Laplacian generates a bounded analytic semigroup on where is a bounded domain with smooth boundary in , we obtain the following consequence for our initial problem.
Corollary 3.9. If is a bounded domain with boundary of class in then for each the problem (3.50) is -well posed, that is, has a unique solution
We note that the same assertion remain true for all
The first author is partially supported by CNPQ/Brazil. The second author is partially financed by Laboratorio de Análisis Estocástico, Proyecto Anillo ACT-13.
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Copyright © 2009 Claudio Cuevas and Carlos Lizama. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.